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Properties

Label 2-440-1.1-c1-0-7
Degree 22
Conductor 440440
Sign 11
Analytic cond. 3.513413.51341
Root an. cond. 1.874411.87441
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 5-s + 7-s + 6·9-s − 11-s − 6·13-s + 3·15-s + 3·17-s − 5·19-s + 3·21-s − 2·23-s + 25-s + 9·27-s − 5·29-s + 5·31-s − 3·33-s + 35-s − 37-s − 18·39-s − 2·41-s + 12·43-s + 6·45-s − 2·47-s − 6·49-s + 9·51-s − 13·53-s − 55-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.447·5-s + 0.377·7-s + 2·9-s − 0.301·11-s − 1.66·13-s + 0.774·15-s + 0.727·17-s − 1.14·19-s + 0.654·21-s − 0.417·23-s + 1/5·25-s + 1.73·27-s − 0.928·29-s + 0.898·31-s − 0.522·33-s + 0.169·35-s − 0.164·37-s − 2.88·39-s − 0.312·41-s + 1.82·43-s + 0.894·45-s − 0.291·47-s − 6/7·49-s + 1.26·51-s − 1.78·53-s − 0.134·55-s + ⋯

Functional equation

Λ(s)=(440s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(440s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 440440    =    23⋅5⋅112^{3} \cdot 5 \cdot 11
Sign: 11
Analytic conductor: 3.513413.51341
Root analytic conductor: 1.874411.87441
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 440, ( :1/2), 1)(2,\ 440,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.4910105322.491010532
L(12)L(\frac12) ≈\approx 2.4910105322.491010532
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1−T 1 - T
11 1+T 1 + T
good3 1−pT+pT2 1 - p T + p T^{2}
7 1−T+pT2 1 - T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1+5T+pT2 1 + 5 T + p T^{2}
23 1+2T+pT2 1 + 2 T + p T^{2}
29 1+5T+pT2 1 + 5 T + p T^{2}
31 1−5T+pT2 1 - 5 T + p T^{2}
37 1+T+pT2 1 + T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1−12T+pT2 1 - 12 T + p T^{2}
47 1+2T+pT2 1 + 2 T + p T^{2}
53 1+13T+pT2 1 + 13 T + p T^{2}
59 1−2T+pT2 1 - 2 T + p T^{2}
61 1−T+pT2 1 - T + p T^{2}
67 1−16T+pT2 1 - 16 T + p T^{2}
71 1−15T+pT2 1 - 15 T + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1−2T+pT2 1 - 2 T + p T^{2}
83 1+14T+pT2 1 + 14 T + p T^{2}
89 1−9T+pT2 1 - 9 T + p T^{2}
97 1+16T+pT2 1 + 16 T + p T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.89766511338015233329150568707, −9.786029738762734281399683357155, −9.483006760033315076271652715999, −8.241837378038505346249898701453, −7.79929638968846771610897877135, −6.73950043879569860448501959504, −5.18236780553163224027614577017, −4.07821246743313595622649858247, −2.77079934605572561635474291231, −1.97350758509779452319901265349, 1.97350758509779452319901265349, 2.77079934605572561635474291231, 4.07821246743313595622649858247, 5.18236780553163224027614577017, 6.73950043879569860448501959504, 7.79929638968846771610897877135, 8.241837378038505346249898701453, 9.483006760033315076271652715999, 9.786029738762734281399683357155, 10.89766511338015233329150568707

Graph of the ZZ-function along the critical line